Reamers in which the position of the tool bit can be adjusted in the radial direction are described, for example, in U.S. Pat. No. 2,977,829. There, a cutting disk in the shape of a quadrilateral is arranged in a recess in the cutter head and is pressed against the shoulder of the cutter head by means of a clamping shoe. Two movable adjustment members positioned in two bores of the cutter head are provided for radial adjustment of the position of the tool bit and for adjustment of the position of the auxiliary cutting edge which is tapered in the direction towards the shaft. The adjustment members engage in the recess of the cutter head and abut the surface opposite the auxiliary cutting edge then being used.